A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation

Guoguang Lin; Yuhang Chen

doi:doi:10.11648/j.ajam.20200806.12

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A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation

Guoguang Lin, Yuhang Chen

Published in American Journal of Applied Mathematics (Volume 8, Issue 6)

Received: 4 November 2020 Accepted: 16 November 2020 Published: 24 November 2020

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Abstract

In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 6)
DOI	10.11648/j.ajam.20200806.12
Page(s)	300-310
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Kirchhoff-Type Equation, Prior Estimation, Galerkin Method, A family of Global Attractors, Hausdorff Dimension, Fractal Dimension

References

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[2]	Guoguang Lin. Nonlinear evolution equation [M]. Kunming: Yunnan University Press, 2011.
[3]	Xilai Dai, Guoguang Lin. Existence and uniqueness of solutions to strongly damped wave equations [J]. Journal of Yunnan University: Natural Science Edition, 2011, 33 (S2): 327-332.
[4]	Narasimha K. Nonlinear vibration of an elastic string [J]. Sound Vibration, 1968, 8: 134-146.
[5]	Boling Guo. Nonlinear evolution equation [M]. Shanghai: Shanghai Science and Technology Press, 1995.
[6]	Zhengde Dai. Zhengde Dai Papers Collection [C]. Kunming: Yunnan University Press, 2016.
[7]	Mitsuhiro N. An attractor for a nonlinear dissipative wave equation of Kirchhoff type [J]. Journal of Mathematical Analysis and Applications, 2009, 353 (2): 652-659.
[8]	Zaiyun Zhang, Zhenhai Liu, Xiujin Miao. Estimate on the Dimension of Global Attractor for Nonlinear Dissipative Kirchhoff Equation [J]. Acta Applicandae Mathematicae, 2010, 110 (1): 271-282.
[9]	Igor C. Longtime dynamics of Kirchhoff wave models with strong nonlinear Damping [J]. Journal of Differen- tial Equations, 2012, 252 (2): 1229-1262.
[10]	Yunlong Gao, Yuting Sun, Guoguang Lin. The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higer-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping [J]. International Journal of Modern Nonlinear Theory and Application 2016, 5 (4): 185-202.
[11]	Matsuyama T, Ikerata R. On the global solutions and energy decay for the wave equations of Kirchhoff-type with nonlinear damping terms [J]. Journal of Mathematical Analysis and Applications, 1996, 204 (3): 729-753.
[12]	Kajitani K, Satoh A. On extension of solutions of Kirchhoff equations [J]. Journal of the Mathematical Society of Japan, 2004, 56 (2): 405-416.
[13]	Lei Wan, Jinbao Dang, Guoguang Lin. Global attractor and dimension estimation of Fractional nonlinear Schrodinge equation [J]. Journal of Yunnan University: Natural Science Edition, 2010, 32 (2): 130-135.
[14]	Ling Chen, Wei Wang, Guoguang Lin. The global attractors and the Hausdoff and fractal demensions estimation for the higher-order nonlinear Kirchhoff-type Equation [J]. Journal of Advances in Mathematics. 2016, 12 (09): 6608-6621.
[15]	Guoguang Lin, Sanmei Yang. Hausdoff and Fractal demensions of the global attractor for the higher-order coupled Kirchhoff-type Equation [J]. Journal of Applied Mathematics and Physics. 2017, 5: 2411-2424.

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APA Style

Guoguang Lin, Yuhang Chen. (2020). A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. American Journal of Applied Mathematics, 8(6), 300-310. https://doi.org/10.11648/j.ajam.20200806.12

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ACS Style

Guoguang Lin; Yuhang Chen. A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. Am. J. Appl. Math. 2020, 8(6), 300-310. doi: 10.11648/j.ajam.20200806.12

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AMA Style

Guoguang Lin, Yuhang Chen. A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. Am J Appl Math. 2020;8(6):300-310. doi: 10.11648/j.ajam.20200806.12

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@article{10.11648/j.ajam.20200806.12,
  author = {Guoguang Lin and Yuhang Chen},
  title = {A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {300-310},
  doi = {10.11648/j.ajam.20200806.12},
  url = {https://doi.org/10.11648/j.ajam.20200806.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.12},
  abstract = {In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.},
 year = {2020}
}

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TY  - JOUR
T1  - A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation
AU  - Guoguang Lin
AU  - Yuhang Chen
Y1  - 2020/11/24
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200806.12
DO  - 10.11648/j.ajam.20200806.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 300
EP  - 310
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200806.12
AB  - In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.
VL  - 8
IS  - 6
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Author Information

Guoguang Lin

School of Mathematics and Statistics, Yunnan University, Kunming, China
Yuhang Chen

School of Mathematics and Statistics, Yunnan University, Kunming, China

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APA Style

Guoguang Lin, Yuhang Chen. (2020). A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. American Journal of Applied Mathematics, 8(6), 300-310. https://doi.org/10.11648/j.ajam.20200806.12

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ACS Style

Guoguang Lin; Yuhang Chen. A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. Am. J. Appl. Math. 2020, 8(6), 300-310. doi: 10.11648/j.ajam.20200806.12

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AMA Style

Guoguang Lin, Yuhang Chen. A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. Am J Appl Math. 2020;8(6):300-310. doi: 10.11648/j.ajam.20200806.12

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@article{10.11648/j.ajam.20200806.12,
  author = {Guoguang Lin and Yuhang Chen},
  title = {A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {300-310},
  doi = {10.11648/j.ajam.20200806.12},
  url = {https://doi.org/10.11648/j.ajam.20200806.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.12},
  abstract = {In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation
AU  - Guoguang Lin
AU  - Yuhang Chen
Y1  - 2020/11/24
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200806.12
DO  - 10.11648/j.ajam.20200806.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 300
EP  - 310
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200806.12
AB  - In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.
VL  - 8
IS  - 6
ER  -

Copy | Download